Step-by-step Solution:

We are given two conditions involving sets \( A \), \( B \), and \( X \):
1. \( A \cap X = B \cap X = \emptyset \)
This means that neither \( A \) nor \( B \) has any elements in common with \( X \).
2. \( A \cup X = B \cup X \)
This means that the union of \( A \) with \( X \) is the same as the union of \( B \) with \( X \).
Step 1: Express \( A \) and \( B \) in Terms of \( X \)
From the second condition: \[ A \cup X = B \cup X \] Since we know that \( X \) is disjoint from both \( A \) and \( B \), the above equation implies: \[ A \setminus X = B \setminus X \] Since \( A \cap X = \emptyset \) and \( B \cap X = \emptyset \), we can write: \[ A = (A \setminus X), \quad B = (B \setminus X) \] Thus, the equation \( A \setminus X = B \setminus X \) simplifies to: \[ A = B \] Conclusion:
From the given conditions, we conclude that:
\[ A = B \] Thus, the sets \( A \) and \( B \) are equal. The required relation between \( A \) and \( B \) is: \[ {A = B} \]